outer measure
Definition [1, 2, 3] Let $X$ be a set, and let $\mathcal{P}(X)$ be the power set^{} of $X$. An outer measure^{} on $X$ is a function ${\mu}^{\ast}:\mathcal{P}(X)\to [0,\mathrm{\infty}]$ satisfying the properties

1.
${\mu}^{\ast}(\mathrm{\varnothing})=0$.

2.
If $A\subset B$ are subsets in $X$, then ${\mu}^{\ast}(A)\le {\mu}^{\ast}(B)$.

3.
If $\{{A}_{i}\}$ is a countable^{} collection^{} of subsets of $X$, then
$${\mu}^{\ast}(\bigcup _{i}{A}_{i})\le \sum _{i}{\mu}^{\ast}({A}_{i}).$$
Here, we can make two remarks. First, from (1) and (2), it follows that ${\mu}^{\ast}$ is a positive function on $\mathcal{P}(X)$. Second, property (3) also holds for any finite collection of subsets since we can always append an infinite^{} sequence of empty sets^{} to such a collection.
References
 1 A. Mukherjea, K. Pothoven, Real and Functional analysis^{}, Plenum press, 1978.
 2 A. Friedman, Foundations of Modern Analysis^{}, Dover publications, 1982.
 3 G.B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999.
Title  outer measure 

Canonical name  OuterMeasure 
Date of creation  20130322 13:45:20 
Last modified on  20130322 13:45:20 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  6 
Author  mathcam (2727) 
Entry type  Definition 
Classification  msc 60A10 
Classification  msc 28A10 
Related topic  CaratheodorysExtensionTheorem 
Related topic  CaratheodorysLemma 
Related topic  ProofOfCaratheodorysExtensionTheorem 